Finding the Other Zeroes of a Cubic Polynomial

Video Explanation

Question

If \(4\) is a zero of the cubic polynomial

\[ f(x) = x^3 – 3x^2 – 10x + 24, \]

find its other two zeroes.

Solution

Step 1: Verify the Given Zero

Substitute \(x = 4\) in \(f(x)\):

\[ f(4) = (4)^3 – 3(4)^2 – 10(4) + 24 \]

\[ = 64 – 48 – 40 + 24 = 0 \]

Hence, \(4\) is a zero of the polynomial.

Step 2: Divide the Polynomial by \((x – 4)\)

Since \(4\) is a zero, \((x – 4)\) is a factor of the polynomial.

Dividing \(x^3 – 3x^2 – 10x + 24\) by \((x – 4)\), we get:

\[ x^3 – 3x^2 – 10x + 24 = (x – 4)(x^2 + x – 6) \]

Step 3: Find the Remaining Zeroes

Factorise the quadratic polynomial:

\[ x^2 + x – 6 = (x + 3)(x – 2) \]

So,

\[ x^3 – 3x^2 – 10x + 24 = (x – 4)(x + 3)(x – 2) \]

Step 4: Write All the Zeroes

Equating each factor to zero:

\[ x – 4 = 0 \Rightarrow x = 4 \]

\[ x + 3 = 0 \Rightarrow x = -3 \]

\[ x – 2 = 0 \Rightarrow x = 2 \]

Conclusion

The other two zeroes of the given cubic polynomial are:

\[ \boxed{-3 \text{ and } 2} \]